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Practical representations of probability sets: aguided tour with applications

Sébastien Destercke

in collaboration with E. Miranda, I. Montes, M. Troffaes, D.Dubois, O. Strauss, C. Baudrit, P.H. Wuillemin.

CNRS researcher, Laboratoire Heudiasyc, Compiègne

Madrid Seminar

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Introduction Basics Practical Representations Applications

Plan

l Introduction

l Basics of imprecise probabilities

l A tour of practical representations

l Illustrative applications

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Where is Compiegne

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Heudiasyc and LABEX MS2T activities

Heudiasycl ' 140 members

l 6M budgetl 4 teams:

m Uncertainty and machinelearning

m Automatic and roboticm Artificial intelligencem Operational research and

networks

LABEX MS2Tl Topic: systems of systemsl 3 laboratories:

m Heudiasycm BMBI: Bio-mechanicm Roberval: mechanic

If interested in collaborations, letme know

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Talk in a nutshell

What is this talk about1. (very) Basics of imprecise probability

2. A review of practical representations

3. Some applications

What is this talk not aboutl Deep mathematics of imprecise probabilities (you can ask Nacho or

Quique)

l Imprecise parametric models

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Plan

l Introduction




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Imprecise probabilities

What?Representing uncertainty as a convex set P of probabilities rather than asingle one

Why?l precise probabilities inadequate to model lack of information;

l generalize set-uncertainty and probabilistic uncertainty;

l can model situations where probabilistic information is partial;

l allow axiomatically alternatives to possibly be incomparable

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Probabilities

Probability mass on finite space X = {x1, . . . ,xn} equivalent to a ndimensional vector

p := (p(x1), . . . ,p(xn))

Limited to the set PX of all probabilities

p(x)> 0,∑

x∈X

p(x)= 1 and

The set PX is the (n−1)-unit simplex.

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Point in unit simplex

p(x1)= 0.2, p(x2)= 0.5, p(x3)= 0.3

p(x3)

p(x1)

p(x2)

11

1

p(x2)

p(x3) p(x1)

∝ p(X1 )

∝p(x2

)

∝ p(x 3)

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Imprecise probability

Set P defined as a set of n constraints

E(fi)≤∑

x∈X

fi(x)p(x)≤ E(fi)

where fi :→R bounded functions

Example2p(x2)−p(x3)≥ 0

f (x1)= 0, f (x2)= 2, f (x3)=−1,E(f )= 0

Lower/upper probabilities

Bounds P(A),P(A) on event A equivalent to

P(A)≤ ∑x∈A

p(x)≤P(A)

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Set P example

2p(x2)−p(x3)≥ 0

p(x3)

p(x1)

p(x2)

11

1p(x2)

p(x3) p(x1)

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Credal set example

2p(x2)−p(x3)≥ 02p(x1)−p(x2)−p(x3)≥ 0

P

p(x3)

p(x1)

p(x2)

11

1p(x2)

p(x3) p(x1)

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Natural extension

From an initial set P defined by constraints, we can compute

l The lower expectation E(g) of any function g as

E(g)= infp∈P

E(g)

l The lower probability P(A) of any event A as

P(A)= infp∈P

P(A)

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Some usual problems

l Computing E(g)= infp∈P E(g) of new function g

l Updating P (θ|x)= L(x |θ)P (θ)

l Computing conditional E(f |A)l Simulating/sampling P

l Building joint over variables X1, . . . ,Xn

can be difficult to perform in general → practical representations reducecomputational cost

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What makes a representation "practical"

l A reasonable, algorithmically enumerable number of extreme points

reminderp ∈P extreme iff p =λp1 + (1−λ)p2 with λ ∈ (0,1) implies p1 = p2 = p.

We will denote E (P ) the set of extreme points of P

l n-monotone property of P

2-monotonicity (sub-modularity, convexity)

P(A∪B)+P(A∩B)≥P(A)+P(B) for any A,B ⊆X

∞-monotonicity

P(∪ni=1Ai)≥

∑A⊆{A1,...,An}

−1|A|+1P(∪Ai∈A Ai) for any A1, . . . ,An ⊆X and n > 0

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Extreme points: illustration

l p(x1)= 1,p(x2)= 0,p(x3)= 0l p(x1)= 0,p(x2)= 1,p(x3)= 0l p(x1)= 0.25,p(x2)= 0.25,p(x3)= 0.5

p(x2)

p(x3) p(x1)

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Extreme points: utility

l Computing E(g)→ minimal E on ext. points

l Updating → update extreme points, take convex hull

l Conditional E(f |A) → minimal E(f |A) on ext. points

l Simulating P → take convex mixtures of ext. points

l Joint over variables X1, . . . ,Xn → convex hull of joint extreme

Again, if number of extreme points is limited, or inner approximation (bysampling) acceptable.

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2-monotonicity

Computing E(g)Choquet integral

E(g)= infg+∫ supg

infgP({g ≥ t})dt

In finite spaces → sorting n values of g and compute P(A) for n events

Conditioning

P(A|B)= P(A∩B)P(A∩B)+P(Ac ∩B)

And P(A|B) remains 2-monotone (can be used to get E(f |A))

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∞-monotonicity

If P is ∞-monotone, its Möbius inverse m : 2X →R

m(A)= ∑B⊆A

−1|A\B|P(B)

is positive and sums up to one, and is often called belief function

Simulating P

Sampling m and considering the associated set A

Joint model of X1, . . . ,XN

If m1,m2 corresponds to inverses of X1,X2, consider joint m12 s.t.

m12(A×B)=m1(A) ·m2(B)

l still ∞-monotone

l outer-approximate other def. of independence between P1, P2

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2-monotonicity and extreme points [3]

Generating extreme points if P 2-monotone:

1. Pick a permutation σ : [1,n]→ [1,n] of X

2. Consider sets Aσi = {xσ(1), . . . ,xσ(i)}

3. define Pσ({xσ(i)})=P(Aσi )−P(Aσ

i−1) for i = 1, . . . ,n (Aσ0 =;)

4. then Pσ ∈ E (P )

Some commentsl Maximal value of |E (P )| = n!

l We can have Pσ1 =Pσ2 with σ1 6=σ2 → |E (P )| often less than n!

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Example

l X = {x1,x2,x3}

l σ(1)= 2,σ(2)= 3,σ(3)= 1

l Aσ0 =;,Aσ

1 = {x2},Aσ2 = {x2,x3},Aσ

3 =X

l Pσ({xσ(1)})=Pσ({x2})=P({x2})−P(;)=P({x2})

l Pσ({xσ(2)})=Pσ({x3})=P({x2,x3})−P({x2})

l Pσ({xσ(3)})=Pσ({x1})=P(X )−P({x2,x3})= 1−P({x2,x3})

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Plan

l Introduction


l A tour of practical representationsm Basicsm Possibility distributionsm P-boxesm Probability intervalsm Elementary Comparative probabilities


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Introduction Basics Practical Representations ApplicationsBasics Possibility distributions P-boxes Probability intervals Elem. Compa.

Two very basic models

Probability

l P({xi })=P({xi })=P({xi })

l ∞-monotone, n constraints, |E | = 1

Vacuous model PX

Only support X of probability is known

l P(X )= 1

l ∞-monotone, 1 constraints, |E (P )| = n (Dirac distribution)

Easily extends to vacuous on set A (can be used in robust optimisation,decision under risk, interval-analysis)

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A concise graphProbaVacuous

Linear-vacuous Pari-mutuel

Possibilities

P-boxes

Prob. int

Compa.

∞-monotone

2-monotone

Model A

Model B

A special case of B

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Neighbourhood models

Build a neighbourhood around a given probability P0

Linear vacuous/ε-contaminationl P(A)= (1−ε)P0(A)+ (ε)PX (A)

l ∞-monotone, n+1 constraints, |E (P )| = n

l ε ∈ [0,1]: unreliability of information P0

Pari-Mutuel [16]l P(A)=max{(1+ε)P0(A)−ε,0}

l 2-monotone, n+1 constraints, |E (P )| =? (n?)

l ε ∈ [0,1]: unreliability of information P0

Other models exist, such as odds-ratio or distance-based (all q s.t. d(p,q)< δ)→ often not attractive for |E (P )|/monotonicity, but may have nice properties(odds/ratio: updating, square/log distances: convex continuous neighbourhood)

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Illustration

x2

x3 x1

pari-mutuellinear-vacuous

P0 = (0.5,0.3,0.2)

ε= 0.2

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Possibilities

P-boxes

Prob. int

Compa.

∞-monotone

2-monotone

Model A

Model B

A special case of B

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Possibility distributions [10]

DefinitionDistribution π :X → [0,1]

with π(x)= 1 for at least one x

P given by

P(A)= minx∈Ac

1−π(x),

which is a necessity measure

π

Characteristics of Pl Necessitates at most n values

l P is an ∞-monotone measure

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Possibility distributions

Alternative definitionProvide nested events

A1 ⊆ . . . ⊆An.

Give lower confidence bounds

P(Ai)=αi

with αi+1 ≥αi

Ai

αi

Extreme points [19]

l Maximum number is 2n−1

l Algorithm using nested structure of sets Ai

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A basic distribution: simple support

l Set E of most plausiblevalues

l Confidence degree α=P(E)

Extends to multiple sets E1, . . . ,Ep

→ Confidence degrees overnested sets [18]

pH value ∈ [4.5,5.5] with

α= 0.8 (∼ "quite probable")

π

3 4 4.5 5.5 6 70

0.20.40.60.81.0

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Partially specified probabilities [1] [8]

Triangular distribution M , [a,b]encompasses all probabilitieswith

l mode/reference value M

l support domain [a,b].

Getting back to pH

l M = 5

l [a,b]= [3,7]

1

pH

π

5 73

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Normalized likelihood as possibilities [9] [2]

π(θ)= L(θ|x)/maxθ∗∈ΘL(θ∗|x)

Binomial situation:

l θ = success probability

l x number of observedsuccesses

l x= 4 succ. out of 11

l x= 20 succ. out of 55

θ

1π

4/11

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Other examples

l Statistical inequalities (e.g., Chebyshev inequality) [8]

l Linguistic information (fuzzy sets) [5]

l Approaches based on nested models

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Possibilities

P-boxes

Prob. int

Compa.

∞-monotone

2-monotone

Model A

Model B

A special case of B

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P-boxes [6]

DefinitionWhen X ordered, bounds onevents of the kind:

Ai = {x1, . . . ,xi }

Each bounded by

F (xi)≤P(Ai)≤F (xi)

0.5

1.0

x1 x2 x3 x4 x5 x6 x7

Characteristics of Pl Necessitates at most 2n values

l P is an ∞-monotone measure

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In general

DefinitionA set of nested events

A1 ⊆ . . . ⊆An

Each bounded by

αi ≤P(Ai)≤βi

0.5

1.0

x1 x2 x3 x4 x5 x6 x7

Extreme points [15]l At most equal to Pell number Kn = 2Kn−1 +Kn−2

l Algorithm based on a tree structure construction

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P-box on reals [11]

A pair [F ,F ] of cumulativedistributions

Bounds over events [−∞,x ]

l Percentiles by experts;

l Kolmogorov-Smirnov bounds;

Can be extended to anypre-ordered space [6], [21] ⇒multivariate spaces!

Expert providing percentiles

0≤P([−∞,12])≤ 0.2

0.2≤P([−∞,24])≤ 0.4

0.6≤P([−∞,36])≤ 0.8

0.5

1.0

6 12 18 24 30 36 42E1

E2

E3

E4

E5

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Possibilities

P-boxes

Prob. int

Compa.

∞-monotone

2-monotone

Model A

Model B

A special case of B

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Probability intervals [4]

DefinitionElements {x1, . . . ,xn}.

Each bounded by

p(xi) ∈ [p(xi),p(xi)] x1 x2 x3 x4 x5 x6

Characteristics of Pl Necessitates at most 2n values

l P is an 2-monotone measure

Extreme points [4]l Specific algorithm to extract

l If n even, maximum number is(n+1

n/2

)n2

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Probability intervals: example

Linguistic assessmentl x is very probable

l x has a good chance

l x is very unlikely

l x probability is about α

⇒Numerical translation

l p(x)≥ 0.75

l 0.4≤ p(x)≤ 0.85

l p(x)≤ 0.25

l α−0.1≤ p(x)≤α+1

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Possibilities

P-boxes

Prob. int

Compa.

∞-monotone

2-monotone

Model A

Model B

A special case of B

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Comparative probabilities

definitionsComparative probabilities on X : assessments

P(A)≥P(B)

event A "at least as probable as" event B.

Some commentsl studied from the axiomatic point of view [13, 20]

l few studies on their numerical aspects [17]

l interesting for qualitative uncertainty modeling/representation,expert elicitation, . . .

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A specific case: elementary comparisons [14]

Elementary comparisonsComparative probability orderings of the states X = {x1, . . . ,xn} in theform of a subset L of {1, . . . ,n}× {1, . . . ,n}.

The set of probability measures compatible with this information is

P (L )= {p ∈PX |∀(i , j) ∈L ,p(xi)≥ p(xj)},

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Why focusing on this case?

Practical interestl multinomial models (e.g., imprecise prior for Dirichlet), modal value

elicitation

l direct extension to define imprecise belief functions

Easy to represent/manipulatel Through a graph G = (X ,L ) with states as nodes and relation L

for edges

l Example: given X = {x1, . . . ,x5},L = {(1,3),(1,4),(2,5),(4,5)}, itsassociated graph G is:

x1

x3 x4

x2

x5

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Some properties

Characteristics of P

l Necessitates at most n2 values

l No guarantee that P is a 2-monotone measure

Extreme points [14]l Algorithm identifying subsets of disconnected nodes

l maximal number is 2n−1

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A concise list (acc. to my knowledge)

Name Monot. Max. |const.| Max. |E (P )| Algo. to get E (P )Proba ∞ n 1 Yes

Vacuous ∞ 1 n Yes2-mon 2 2n n! Yes [3]∞-mon ∞ 2n n! NoLin-vac. ∞ n+1 n Yes

Pari-mutuel 2 n+1 ? (n) NoPossibility ∞ n 2n−1 Yes [19]

P-box (gen.) ∞ 2n Kn (Pell) Yes [15]Prob. int. 2 2n

(n+1n/2

)n2 Yes [4]

Elem. Compa. × n2 2n−1 Yes [14]

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A concise final graphProbaVacuous


Possibilities

P-boxes

Prob. int

Compa.

∞-monotone

2-monotone

Model A

Model B

A special case of B

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Some open questions

l study the numerical aspects of comparative probabilities withnumbers/general events.

l study the potential link between possibilities and elementarycomparative probabilities (share same number of extreme points,induce ordering between states).

l study restricted bounds/information over specific families of events,other than nested/elementary ones (e.g., events of at most k states).

l look at probability sets induced by bounding specific distances to p0,in particular L1,L2,L∞ norms.

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Plan

l Introduction



l Illustrative applicationsm Numerical Signal processing [7]m Camembert ripening [12]

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Introduction Basics Practical Representations ApplicationsNumerical filtering Camembert ripening

Signal processing: introduction

l Impulse response µ

filter

µ

l Filtering: convolving kernel µ and observed output f (x)

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Link with probability

l If µ positive and∫Rµ(x) dx = 1

l µ equivalent to probability densityl Convolution: compute mathematical expectation Eµ(f )l Numerical filtering: discretize (sampling) µ and f

f f

x0

µ

x0

µ

l µ> 0,∑

xi µ(xi)= 1

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Which bandwidth?

Rx

µ

∆???

→ use imprecise probabilistic models to model sets of bandwidth→ possiblities/p-boxes with sets centred around x

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Example on simulated signal

time (msec)

signa

l am

plitu

de maxitive upper enveloppe

maxitive lower enveloppe

cloudy upper enveloppe

cloudy lower enveloppe

original signal

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Application: pepper/salt noise removal

Original image Noisy image

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Results

Zoom on 2 parts

Résultats

CWMF ROAD Us

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Motivations

A complex system

The Camembert-type cheese ripening process

tj1 j14Cheese making Cheese ripening (∼ 13◦C, ∼ 95%hum) Warehouse (∼ 4◦C)

P (t = 14

|t = 0

)l Multi-scale modeling; from microbial activities to sensory propertiesl Dynamic probabilistic modell Knowledge is fragmented, heterogeneous and incompletel Difficulties to learn precise model parameters

Use of ε-contamination for a robustness analysis of the model

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Experiments

The network

T (t)

Km(t)

lo(t)

Km(t+1)

lo(t+1)

Time slice t Time slice t +1

Unrolled over 14 time steps (days)

T (1)

Km(1)

lo(1)

T (2)

Km(2)

lo(2)

Km(14)

lo(14)

tj1 j14Cheese making Cheese ripening (∼ 13◦C, ∼ 95%hum) Warehouse (∼ 4◦C)

...

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Propagation resultsForward propagation, ∀t ∈ �1,τ�,T (t)= 12oC (average ripening room temperature) :

Ext{Km(t)|{Km(1), lo(1),T (1), . . . ,T (τ)}}Ext{lo(t)|{Km(1), lo(1),T (1), . . . ,T (τ)}}

no physical constraints with added physical constraints

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Conclusions

Use of practical representationsl +: "Easy" robustness analysis of precise methods, or approximation

of imprecise ones

l +: allow experts to express imprecision or partial information

l +: often easier to explain/represent than general ones

l -: usually focus on specific events

l -: their form may not be conserved by information processing

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References I

[1] C. Baudrit and D. Dubois.Practical representations of incomplete probabilistic knowledge.Computational Statistics and Data Analysis, 51(1):86–108, 2006.

[2] M. Cattaneo.Likelihood-based statistical decisions.In Proc. 4th International Symposium on Imprecise Probabilities andTheir Applications, pages 107–116, 2005.

[3] A. Chateauneuf and J.-Y. Jaffray.Some characterizations of lower probabilities and other monotonecapacities through the use of Mobius inversion.Mathematical Social Sciences, 17(3):263–283, 1989.

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References II

[4] L. de Campos, J. Huete, and S. Moral.Probability intervals: a tool for uncertain reasoning.I. J. of Uncertainty, Fuzziness and Knowledge-Based Systems,2:167–196, 1994.

[5] G. de Cooman and P. Walley.A possibilistic hierarchical model for behaviour under uncertainty.Theory and Decision, 52:327–374, 2002.

[6] S. Destercke, D. Dubois, and E. Chojnacki.Unifying practical uncertainty representations: I generalizedp-boxes.Int. J. of Approximate Reasoning, 49:649–663, 2008.

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References III

[7] S. Destercke and O. Strauss.Filtering with clouds.Soft Computing, 16(5):821–831, 2012.

[8] D. Dubois, L. Foulloy, G. Mauris, and H. Prade.Probability-possibility transformations, triangular fuzzy sets, andprobabilistic inequalities.Reliable Computing, 10:273–297, 2004.

[9] D. Dubois, S. Moral, and H. Prade.A semantics for possibility theory based on likelihoods,.Journal of Mathematical Analysis and Applications, 205(2):359 –380, 1997.

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References IV

[10] D. Dubois and H. Prade.Practical methods for constructing possibility distributions.International Journal of Intelligent Systems, 31(3):215–239, 2016.

[11] S. Ferson, L. Ginzburg, V. Kreinovich, D. Myers, and K. Sentz.Constructing probability boxes and dempster-shafer structures.Technical report, Sandia National Laboratories, 2003.

[12] M. Hourbracq, C. Baudrit, P.-H. Wuillemin, and S. Destercke.Dynamic credal networks: introduction and use in robustnessanalysis.In Proceedings of the Eighth International Symposium on ImpreciseProbability: Theories and Applications, pages 159–169, 2013.

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References V

[13] B. O. Koopman.The axioms and algebra of intuitive probability.Annals of Mathematics, pages 269–292, 1940.

[14] E. Miranda and S. Destercke.Extreme points of the credal sets generated by comparativeprobabilities.Journal of Mathematical Psychology, 64:44–57, 2015.

[15] I. Montes and S. Destercke.On extreme points of p-boxes and belief functions.In Int. Conf. on Soft Methods in Probability and Statistics (SMPS),2016.

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References VI

[16] R. Pelessoni, P. Vicig, and M. Zaffalon.Inference and risk measurement with the pari-mutuel model.International journal of approximate reasoning, 51(9):1145–1158,2010.

[17] G. Regoli.Comparative probability orderings.Technical report, Society for Imprecise Probabilities: Theories andApplications, 1999.

[18] S. Sandri, D. Dubois, and H. Kalfsbeek.Elicitation, assessment and pooling of expert judgments usingpossibility theory.IEEE Trans. on Fuzzy Systems, 3(3):313–335, August 1995.

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References VII

[19] G. Schollmeyer.On the number and characterization of the extreme points of thecore of necessity measures on finite spaces.ISIPTA conference, 2016.

[20] P. Suppes, G. Wright, and P. Ayton.Qualitative theory of subjective probability.Subjective probability, pages 17–38, 1994.

[21] M. C. M. Troffaes and S. Destercke.Probability boxes on totally preordered spaces for multivariatemodelling.Int. J. Approx. Reasoning, 52(6):767–791, 2011.

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